Express 0.275 as a fraction in its lowest term.

Midnight can be considered 00:00 (start of the day) or 24:00 (end of the day).

• Mafeking: 14:40. Difference from 24:00 is $24 \text{ hours} - 14 \text{ hours } 40 \text{ minutes} = 9 \text{ hours } 20 \text{ minutes}$.
• Gaborone: 19:45. Difference from 24:00 is $24 \text{ hours} - 19 \text{ hours } 45 \text{ minutes} = 4 \text{ hours } 15 \text{ minutes}$.
• Gaborone: 19:55. Difference from 24:00 is $24 \text{ hours} - 19 \text{ hours } 55 \text{ minutes} = 4 \text{ hours } 5 \text{ minutes}$.
• Francistown: 07:13. Difference from 00:00 is $7 \text{ hours } 13 \text{ minutes}$.
• Francistown: 07:23. Difference from 00:00 is $7 \text{ hours } 23 \text{ minutes}$.
• Plumtree: 09:55. Difference from 00:00 is $9 \text{ hours } 55 \text{ minutes}$.
• Plumtree: 10:05. Difference from 00:00 is $10 \text{ hours } 5 \text{ minutes}$.
• Bulawayo: 12:45. Difference from 00:00 is $12 \text{ hours } 45 \text{ minutes}$.

Comparing the differences:

• Gaborone (19:55) has the smallest difference of 4 hours 5 minutes.
• Bulawayo (12:45) has a difference of 12 hours 45 minutes.

Therefore, the train from Gaborone (19:55) is nearest to midnight.

The train nearest to midnight is:

$$\boxed{\text{Gaborone}}$$